This result allows us to further improve the theoretical
bounds from [1]} on the parameter
\(\sigma \) . Namely, it has been previously established that
\(\sigma \)  grows at least linearly with the increasing subset
size of \(\tau \) -uniform sampling, i.e., \(\tau \sigma (1) \le \sigma (\tau )\) . We can establish more informative bounds
depending on the eigenvalue decay. Specifically, for a
decreasing sequence of eigenvalues \(\lbrace \lambda _i\rbrace _{i=1}^d\) ,
\(\left(1 + \sum _{j=1}^{\tau -1}\frac{\lambda _j}{\lambda _d}\right)\sigma (1) \le \sigma (\tau ).\) 
